A stable fixed focus and its immunity zone in a second-order controlled dynamical system (Q2388055)

Consider the system \[ \dot x=P(x)+u(t)Q(x), \tag{1} \] where \(x=(x_1, x_2)\), \(P(x)=(P(x_1),P(x_2))\) and \(Q(x)=(Q(x_1),Q(x_2))\) are vector functions in \(C^k\), \(k\geq 5\), the control \(u:\mathbb{R}^1\to \mathbb{R}^1\) is a piecewise continuous bounded function on each finite interval, and \(m\leq u(t)\leq n\). Denote (1) with \(u(t)=\mu =const\), \(m\leq\mu\leq n\) as (2). Suppose (2) has a stable fixed focus with \(P(0)=Q(0)=0\). Then \(0\) coincides with a singular point of the contact curve \(F(x)=\) det\((P(x),Q(x))=f_1(x)f_2(x)=0\) satisfying \(f_1(0)f_2(0)=0\) and nonzero Jacobian in the neighborhood \(H_{\varepsilon}(0)\) of \(0\). The immunity zone \(I(0)\) of the state \(0\) is the maximum safe zone \(B(0)\). The set \(B(0)\) is a subset of the controllability set \(U(0)\) in the state \(0\) of (2) containing only one stable limit set \(0\). \(B(0)\) satisfies \(D(M_o)\subset B(0)\) for each point \(M_o\in B(0)\), where \(D(M_o)\) is the attainability set from \(M_o\). The sewed \(mn\,(nm)\)-system is a system (1) with \(u(t)=m(n)\;[u(t)=n(m)]\) in the part of the plane in which \(F(x)>0\;[F(x)<0]\), resp., and \(m\leq u(t)\leq n\) for \(F(x)=0\). The character of the equilibrium of \(0\) of the \(nm\)-system is analyzed using the Poincaré function determined by the trajectories of the sewed system. The paper presents the result that the constrained control of (1) belongs to the stability domain of \(0\), but the controllability set in \(H_{\varepsilon}(0)\) does not include safe zones.